Series of families of degree six circulant graphs

An approach for constructing and optimizing graphs of series of analytically described circulant graphs of degree six with general topological properties is proposed. The paper presents three series of families of undirected circulants having the form $C(N(d,p); 1, s_2(d,p), s_3(d,p))$, with an arbitrary diameter $d>1$ and a variable parameter $p(d)$, $1\le p(d)\le d$. The orders $N$ of each graph in the families are determined by a cubic polynomial function of the diameter, and generators $s_2$ and $s_3$ are defined by polynomials of the diameter of various orders. We have proved that the found series of families include degree six extremal circulant graphs with the largest known orders for all diameters. By specifying the functions $p(d)$, new infinite families of circulant graphs including solutions close to extremal graphs are obtained.